L(s) = 1 | + (−0.992 + 0.120i)2-s + (0.748 + 0.663i)3-s + (0.970 − 0.239i)4-s + (0.464 − 0.885i)5-s + (−0.822 − 0.568i)6-s + (−0.935 + 0.354i)8-s + (0.120 + 0.992i)9-s + (−0.354 + 0.935i)10-s + (−0.992 − 0.120i)11-s + (0.885 + 0.464i)12-s + (0.935 − 0.354i)15-s + (0.885 − 0.464i)16-s + (−0.354 − 0.935i)17-s + (−0.239 − 0.970i)18-s − i·19-s + (0.239 − 0.970i)20-s + ⋯ |
L(s) = 1 | + (−0.992 + 0.120i)2-s + (0.748 + 0.663i)3-s + (0.970 − 0.239i)4-s + (0.464 − 0.885i)5-s + (−0.822 − 0.568i)6-s + (−0.935 + 0.354i)8-s + (0.120 + 0.992i)9-s + (−0.354 + 0.935i)10-s + (−0.992 − 0.120i)11-s + (0.885 + 0.464i)12-s + (0.935 − 0.354i)15-s + (0.885 − 0.464i)16-s + (−0.354 − 0.935i)17-s + (−0.239 − 0.970i)18-s − i·19-s + (0.239 − 0.970i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3699558847 - 0.5152185324i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3699558847 - 0.5152185324i\) |
\(L(1)\) |
\(\approx\) |
\(0.7515895712 - 0.03793168010i\) |
\(L(1)\) |
\(\approx\) |
\(0.7515895712 - 0.03793168010i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.992 + 0.120i)T \) |
| 3 | \( 1 + (0.748 + 0.663i)T \) |
| 5 | \( 1 + (0.464 - 0.885i)T \) |
| 11 | \( 1 + (-0.992 - 0.120i)T \) |
| 17 | \( 1 + (-0.354 - 0.935i)T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (0.120 + 0.992i)T \) |
| 31 | \( 1 + (-0.822 - 0.568i)T \) |
| 37 | \( 1 + (-0.822 - 0.568i)T \) |
| 41 | \( 1 + (-0.663 + 0.748i)T \) |
| 43 | \( 1 + (-0.568 - 0.822i)T \) |
| 47 | \( 1 + (0.239 - 0.970i)T \) |
| 53 | \( 1 + (-0.354 - 0.935i)T \) |
| 59 | \( 1 + (-0.464 + 0.885i)T \) |
| 61 | \( 1 + (0.354 - 0.935i)T \) |
| 67 | \( 1 + (0.239 - 0.970i)T \) |
| 71 | \( 1 + (0.663 - 0.748i)T \) |
| 73 | \( 1 + (0.992 + 0.120i)T \) |
| 79 | \( 1 + (-0.970 - 0.239i)T \) |
| 83 | \( 1 + (0.663 + 0.748i)T \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 + (-0.464 - 0.885i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.19890354061479552452763305000, −20.52041945725957158246852667090, −19.77507063843522155784545733161, −18.79068267274476193840966382548, −18.67140153934384423896528791353, −17.73732753657626498960288321046, −17.27168593558939123183858956535, −15.97991600055253037750327993190, −15.284882661398333246448850569279, −14.54897989742448290114450073561, −13.72186135159247456393234291565, −12.79831009546535903617040956526, −12.075512216994357920570484296270, −11.00924156099448002409166703860, −10.207793177174674364177484678352, −9.71556512483200725823986638325, −8.56814461946685728334224866223, −7.96787258926122895347710840811, −7.26541056437945549327199634808, −6.39623137366889432269687335397, −5.748960601426002668923052587456, −3.8408724910374619923882842440, −2.99381060571084526073331019268, −2.14874589624421385123818087480, −1.551093843756094651537575217470,
0.28307378265212328995179406809, 1.83405815736306484779427908325, 2.47369434026345919978093872945, 3.55216231386730880910343202329, 4.958006736836866386593688471058, 5.38614080904576639963678739675, 6.74543334538927421278828888326, 7.712875610084692763109567792749, 8.44141460710115051392992946267, 9.08934816438055863385085624763, 9.71395484785633355948349119517, 10.465931544170742953052674539656, 11.25191440913972715323598449238, 12.35105287588721854049489227787, 13.35014913645501620369436427715, 13.99449833949294626798395918980, 15.110662685608932408994896081950, 15.80118059989742892194523042683, 16.291069722941138939784925976535, 17.023366111514522841933371474077, 18.04482287327746585433052367819, 18.519960687692672677502738860288, 19.78790761264126046487204618238, 20.07804339355758444933867361199, 20.781837310684465809643501371910